Rectangular Prism Calculator - Volume, Area, and Diagonal
Use this rectangular prism calculator to find the volume, surface area, lateral area, and space diagonal of any box, cuboid, or container from three side lengths.
Rectangular Prism Calculator
Results
What Is Rectangular Prism Calculator?
A rectangular prism calculator takes the three perpendicular side lengths of any box, cuboid, or container and returns four useful measurements: the volume V = l * w * h, the surface area SA = 2(lw + lh + wh), the lateral surface area, and the space diagonal d = sqrt(l^2 + w^2 + h^2).
- • Aquarium and fish tank capacity: A 36 x 13 x 16 inch long tank needs both the volume in cubic inches for water capacity and the surface area in square inches for glass.
- • Raised bed and planter soil volume: A 4 x 2 x 0.5 foot raised bed needs the volume in cubic feet for the soil order and the lateral surface area in square feet for the wood panel cuts.
- • Shipping carton and freight measurements: A 24 x 18 x 12 inch carton needs the volume for the billable weight and the space diagonal for the longest object that fits through the diagonal.
A rectangular prism is a three-dimensional solid bounded by six rectangles, with opposite faces equal and parallel. It has 8 vertices, 12 edges, and 6 faces, and the three edges meeting at any vertex are perpendicular. A cube is the special case where all three side lengths are equal.
When only the volume matters, the Rectangular Prism Volume Calculator returns just V = l * w * h alongside the base area for users who do not need the surface area or space diagonal.
How Rectangular Prism Calculator Works
The calculator applies four formulas to the same three perpendicular side lengths: V = l * w * h for the volume, SA = 2(lw + lh + wh) for the surface area, LSA = 2h(l + w) for the lateral surface area, and d = sqrt(l^2 + w^2 + h^2) for the space diagonal.
- l, w: length and width of the rectangular base, in the same linear unit
- h: perpendicular height from the base plane to the opposite face
- V: volume in cubic units, equal to l * w * h
- SA: surface area in square units, equal to 2 * (l*w + l*h + w*h)
- LSA: lateral surface area in square units, equal to 2 * h * (l + w)
- d: space diagonal in linear units, equal to sqrt(l^2 + w^2 + h^2)
The V = l * w * h rule works for any rectangular prism regardless of which face is the base, because multiplication is commutative. The surface area is obtained by pairing opposite faces and doubling the three unique face areas. For a cube the formulas reduce to V = s^3, SA = 6s^2, LSA = 4s^2, and d = s*sqrt(3).
Example with length 12, width 10, and height 8
Enter length = 12, width = 10, height = 8.
V = 12 * 10 * 8 = 960.00 cubic units. SA = 2 * (120 + 96 + 80) = 592.00 square units. LSA = 2 * 8 * 22 = 352.00 square units. d = sqrt(308) = 17.55 linear units.
Volume = 960.00 cubic units. Surface area = 592.00 square units. Lateral surface area = 352.00 square units. Space diagonal = 17.55 linear units.
A 12 x 10 x 8 inch storage bin holds about 15.7 quarts and the longest rod that fits inside is 17.55 inches.
According to Wikipedia, the volume of a cuboid is the product of its three edge lengths, V = a * b * c, and the surface area is SA = 2 * (ab + ac + bc).
For a step-by-step volume walkthrough on the same three side lengths, the Cuboid Volume Calculator shows the V = l * w * h formula alongside the base area and is the closest single-output peer in the same category.
Key Concepts Explained
These four concepts decide which formula the calculator is using and what each result means for the box you are measuring.
Length, Width, and Height
The three side lengths are measured along three perpendicular directions, so any one of them can be the height as long as the other two are the sides of the rectangular base. A 12 x 10 x 8 box returns 960 cubic units, 592 square units, 352 square units, and a 17.55 unit space diagonal regardless of which face is the base.
Surface Area vs Lateral Surface Area
The full surface area counts all six faces, equal to 2(lw + lh + wh), while the lateral surface area counts only the four side faces, equal to 2h(l + w). Use the lateral area for open-top boxes and the full surface area for closed boxes.
The Space Diagonal Rule
The space diagonal is the corner-to-corner line that runs through the inside of the prism. It is the three-dimensional form of the Pythagorean theorem, d = sqrt(l^2 + w^2 + h^2), and it answers how long a rod will just fit inside the box.
Cuboid vs Cube vs Box
A cuboid and a rectangular prism are the same shape, so this calculator handles both names. A cube is a cuboid where all three edges are equal, so V = s^3, SA = 6s^2, LSA = 4s^2, and d = s*sqrt(3).
A common error is to use the diagonal of a face in place of one of the side lengths. The face diagonal is the line across the front of the box, not the perpendicular height, and using it will overstate the volume.
For a focused walkthrough of the full surface area formula, the Cuboid Surface Area Calculator returns the same SA = 2(lw + lh + wh) value in square units and pairs it with each face area.
How to Use This Calculator
Type the three perpendicular side lengths of the box into the form, then read the volume, surface area, lateral surface area, and space diagonal from the result panel.
- 1 Pick the linear unit first: Decide whether to type the three side lengths in inches, feet, centimeters, or meters. The calculator labels each result in the matching cubic, square, or linear unit.
- 2 Enter the length of the rectangular base: Type the longest edge of the rectangular base into the Length box. For a 36 x 13 x 16 inch long aquarium, length is 36.
- 3 Enter the width of the rectangular base: Type the shorter edge of the rectangular base into the Width box. For the same aquarium, width is 13.
- 4 Enter the perpendicular height: Type the third edge into the Height box. For the same aquarium, height is 16.
- 5 Read the four results: Use the Volume row for capacity, the Surface Area row for full outside coverage, the Lateral Surface Area row for open-top boxes, and the Space Diagonal row for the longest object that fits corner to corner.
A 36 x 13 x 16 inch long aquarium has volume 7,488 cubic inches (about 32.4 US gallons), full surface area 2,504 square inches, lateral surface area 1,568 square inches, and a space diagonal of 41.48 inches.
Once the volume is in cubic inches, the Volume Calculator moves the result into gallons, liters, or cubic feet for the fish store, the soil bag, or the shipping manifest.
Benefits of Using This Calculator
A rectangular prism calculator that returns four different measurements from the same three inputs turns one spec sheet into a complete plan for capacity, materials, and longest-fit dimensions.
- • Four results from three inputs: Volume, surface area, lateral surface area, and space diagonal are all returned from the same length, width, and height, with no need to retype the dimensions.
- • Lateral area for open-top boxes: The four side faces only are returned as a separate row, the right number for open-top storage bins, raised beds, and planter boxes.
- • Space diagonal for longest-fit objects: The space diagonal is returned in the chosen linear unit, so it answers how long a rod, panel, or piece of furniture will fit through the box corner to corner.
- • Single unit for every result: Volume uses cubic units, surface area and lateral area use square units, and the space diagonal uses linear units, all derived from the same base unit.
The calculator exposes both V = l * w * h and SA = 2(lw + lh + wh) as separate result rows so the user can verify the formula step without retyping the values.
For a separate surface-area calculation on a different 3D shape, the Surface Area Calculator handles cubes, prisms, cylinders, cones, pyramids, and spheres in one place.
Factors That Affect Your Results
A rectangular prism calculator is a small set of products and sums of three side lengths, but a few measurement choices decide whether the result matches the real box.
Unit consistency
Every length input must use the same unit. Mixing inches and feet, or feet and meters, will produce an answer that is off by a power of 12 or by a factor of about 3.281.
Perpendicular height
The three side lengths must be measured along three perpendicular directions, not along the diagonal of a face. The face diagonal is always longer than the perpendicular height.
Closed box vs open-top box
The full surface area assumes a closed rectangular prism with six faces. An open-top box has the same lateral surface area and the same volume, but the full surface area should drop one of the length-by-width rectangles.
Space diagonal use cases
The space diagonal is the longest line that fits inside the box, the right number for the longest rod, pole, or piece of furniture that will pass through the prism, but not the shipping length on a freight label.
- • The calculator does not solve for a missing side length when only the volume is known, because the same volume can come from many different side-length combinations.
- • Real boxes are rarely perfect rectangular prisms: shipping cartons have rounded corners, plastic bins taper, and aquarium glass has thickness.
For an irregular base that is not a rectangle, the V = l * w * h rule does not apply directly. The base area must be computed for the actual footprint, and the V = B * h rule from any right prism can then be used.
According to Wolfram MathWorld, the space diagonal of a cuboid of edge lengths a, b, c has length d = sqrt(a^2 + b^2 + c^2), and the surface area is 2(ab + bc + ac).
According to Wolfram MathWorld, a right prism has volume equal to base area times perpendicular height, V = B * h, and a rectangular prism is the right prism whose base is a rectangle of area l * w.
For a pyramid that sits on the same rectangular base, the Rectangular Pyramid Volume Calculator applies the same base area l * w but multiplies by h and divides by 3 to get V = (1/3) * l * w * h.
Frequently Asked Questions
Q: How do you calculate the volume and surface area of a rectangular prism?
A: The volume is V = l * w * h, the product of the three perpendicular side lengths. The surface area is SA = 2(lw + lh + wh), which doubles the three unique face areas across the six faces. This calculator returns both results in matching units from the same three inputs.
Q: What is the space diagonal of a rectangular prism?
A: The space diagonal is the corner-to-corner line through the inside of the rectangular prism. Its length is d = sqrt(l^2 + w^2 + h^2), the three-dimensional form of the Pythagorean theorem. It is the longest single line inside the solid.
Q: What units does this rectangular prism calculator return?
A: Use one linear unit for all three side lengths, such as inches, feet, centimeters, or meters. The calculator returns the volume in cubic units, the surface area and lateral surface area in square units, and the space diagonal in linear units.
Q: How is a rectangular prism different from a cube?
A: A cube is a rectangular prism where all three side lengths are equal, so V = s^3, SA = 6s^2, LSA = 4s^2, and d = s * sqrt(3). A rectangular prism is the more general case where the three side lengths can differ, and the same formulas still apply.
Q: What is the lateral surface area of a rectangular prism?
A: The lateral surface area is the area of the four side faces only, excluding the top and bottom. It is LSA = 2h(l + w), the perimeter of the rectangular base multiplied by the height. Use the lateral area for open-top boxes and planters.
Q: How do you find a missing side length of a rectangular prism from the volume?
A: If two side lengths and the volume are known, divide the volume by the product of the two known sides to recover the third side. If only the volume is known, the missing sides are not unique, because many different combinations produce the same product.